A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity

Abstract

Buchdahl transformations provide a geometric route from vacuum spacetimes to scalar-field solutions. We formulate this mechanism as a target-covariant solution generator for Einstein gravity coupled to nonlinear sigma models in $D>3$ dimensions. On any connected region in which a Ricci-flat seed admits a non-null hypersurface-orthogonal cyclic coordinate with fixed-sign nonzero norm, the logarithm of that norm defines a Buchdahl potential $\Sigma$. A one-parameter redistribution of the cyclic norm then generates the exact Ricci source $R_{AB}=C_D(\beta ){\partial }_A\Sigma {\partial }_B\Sigma$, with $C_D(\beta )=\frac{D-2}{D-3}(1-{\beta }^2)$, while $\Sigma$ remains harmonic in the deformed metric. Composing $\Sigma$ with an affinely parametrized geodesic of a non-degenerate target metric whose squared speed is $C_D(\beta )$ therefore yields an exact Einstein--sigma-model solution. For positive-definite targets, the corresponding pullback condition also implies that the nontrivial scalar map has one-dimensional image; for indefinite targets, the geodesic ansatz remains a sufficient solution-generating sector but need not exhaust all possible realizations of the same rank-one spacetime source. Through the Einstein/Jordan-frame correspondence, the construction gives vacuum tensor--multiscalar solutions as well. We illustrate the unified generator with flat, spherical, hyperbolic and axion--dilaton targets and with Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza--Klein bubble and C-metric seeds. The result separates cleanly the geometric Buchdahl source from its target-space realization and extends this rank-one cyclic sector across static, cosmological, wave and accelerating geometries.

23 pages